Differential graded algebras for trivalent plane graphs and their representations
Abstract
To any trivalent plane graph embedded in the sphere, Casals and Murphy associate a differential graded algebra (dgalgebra), in which the underlying graded algebra is free associative over a commutative ring. Our first result is the construction of a generalization of the CasalsMurphy dgalgebra to noncommutative coefficients, for which we prove various functoriality properties not previously verified in the commutative setting. Our second result is to prove that rank $r$ representations of this dgalgebra, over a field $\mathbb{F}$, correspond to colorings of the faces of the graph by elements of the Grassmannian $\operatorname{Gr}(r,2r;\mathbb{F})$ so that bordering faces are transverse, up to the natural action of $\operatorname{PGL}_{2r}(\mathbb{F})$. Underlying the combinatorics, the dgalgebra is a computation of the fully noncommutative Legendrian contact dgalgebra for Legendrian satellites of Legendrian 2weaves, though we do not prove as such in this paper. The graph coloring problem verifies that for Legendrian 2weaves, rank $r$ representations of the Legendrian contact dgalgebra correspond to constructible sheaves of microlocal rank $r$. This is the first such verification of this conjecture for an infinite family of Legendrian surfaces.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.07585
 Bibcode:
 2021arXiv211007585S
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Geometric Topology;
 Mathematics  Symplectic Geometry;
 05C15;
 53D10
 EPrint:
 Minor revisions